When our baby was born (in 2007 Nov.) in Al Ahli hospital, we took the papers to Hamad for registration, they gave us a Registration No., then we took that along with other documents to the NHA building near to Lulu Supermarket/ sports R/A, filled more forms payed the fees and got the birth certificate in a week.
Not sure how is it done now.Maybe the MOI site or the HMC website can give you the latest procedures